Benedict–Webb–Rubin–Starling Equation of State + Hydrate Thermodynamic Theories: An Enhanced Prediction Method for CO₂ Solubility and CO₂ Hydrate Phase Equilibrium in Pure Water/NaCl Aqueous Solution System

Abstract

Accurately predicting the phase behavior and physical properties of carbon dioxide (CO₂) in pure water/NaCl mixtures is crucial for the design and implementation of carbon capture, utilization, and storage (CCUS) technology. However, the prediction task is complicated by CO₂ liquefaction, CO₂ hydrate formation, multicomponent and multiphase coexistence, etc. In this study, an improved method that combines Benedict–Webb–Rubin–Starling equation of state (BWRS EOS) + hydrate thermodynamic theories was proposed to predict CO₂ solubility and phase equilibrium conditions for a mixed system across various temperature and pressure conditions. By modifying the interaction coefficients in BWRS EOS and the Van der Waals–Platteeuw model, this new method is applicable to complex systems containing two liquid phases and a CO₂ hydrate phase, and its high prediction accuracy was verified through a comparative evaluation with a large number of reported experimental data. Furthermore, based on the calculation results, the characteristics of CO₂ solubility and the variation of phase equilibrium conditions of the mixture system were discussed. These findings highlight the influence of hydrates and NaCl on CO₂ solubility characteristics and clearly demonstrate the hindrance of NaCl to the formation of CO₂ hydrates. This study provides valuable insights and fundamental data for designing and implementing CCUS technology that contribute to addressing global climate change and environmental challenges.

1. Introduction

With the intensification of global climate change and the increasing severity of environmental challenges, reducing carbon dioxide (CO₂) emissions has become a focal point of scientific research and industrial sectors [1,2]. Carbon capture, utilization, and storage (CCUS) is a cutting-edge technological solution related to economic development and social progress [3].

The thermodynamic properties of CO₂–pure water/brine mixtures are crucial for designing and implementing CCUS technology, which are influenced by the interactions among CO₂, H₂O, and saline solutes (typically represented by NaCl). The CO₂–pure water/NaCl aqueous solution mixtures involved in CCUS technology span a temperature range from near the freezing point (273.15 K) to room temperature (298.15 K), and a pressure range from atmospheric pressure (101.325 kPa) to the pressure of deep-sea water at an average depth of 3688 m [4] (approximately 40 MPa). The possibly existing phases of the mixtures are CO₂-rich phase (in vapor or liquid state or vapor-liquid coexistence state, contains CO₂ and H₂O, mostly CO₂), H₂O-rich phase (in liquid state, contains CO₂, H₂O, and possibly NaCl, mostly H₂O), and hydrate phase (in solid state, contains CO₂ and H₂O). Accordingly, CO₂ solubility and NaCl fractions, which describe the composition of the fluids in the mixture system and determined the distribution of CO₂-H₂O-NaCl in different phases, directly impacts the performance of CCUS technology.

For instance, CO₂ solubility directly affects the effectiveness, capacity, and long-term stability of CO₂ storage strategies, such as geological sequestration, mineral transformation, and ocean storage [5,6]. The density difference between dissolved CO₂ and other water bodies in natural water systems is a primary factor in determining the mode and velocity of CO₂ migration [7]. In confined environments like rocks or geological formations, this disparity in density significantly influences the pressure distribution and safety of the storage zone. Hydrate-based CO₂ storage technology, an emerging CCUS technology, offers advantages including faster storage rates, higher storage capacity, excellent stability, and cost-effectiveness [8,9,10].

However, the presence of dissolved CO₂ is critically important for the formation of CO₂ hydrate [11]. When CO₂ hydrate is present, CO₂ solubility undergoes a significant decrease with increasing pressure compared to systems without hydrates [12]. Additionally, the solubility is controlled by the equilibrium between the pure water/NaCl aqueous solution phase and the solid CO₂ hydrate phase, whereas in systems without hydrates, the solubility is controlled by the equilibrium between CO₂ and the pure water/NaCl aqueous solution [13]. In the presence of NaCl, the development of hydrogen bonding networks in liquid and hydrate phases are partially impeded [14]. Moreover, water molecules have a higher tendency to adhere to similarly polar salt ions than to nonpolar guest molecules (such as CO₂) [15]. As a result, the thermodynamic properties of CO₂–NaCl aqueous solution mixtures are different from the thermodynamic properties of CO₂–pure water mixtures.

Understanding the phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate is a prerequisite for calculating CO₂ solubility. In the meantime, based on the CO₂ solubility data, the phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate can also be predicted. Limited experimental studies have been conducted on CO₂ solubility and phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution mixtures within the temperature and pressure range of 273.15~298.15 K and 0.1~40 MPa due to the involved extreme environments such as high pressure. The relevant studies we can obtain include: studies on the phase equilibrium conditions of CO₂–pure water [11,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]/NaCl aqueous solution [11,20,30,31,32,34,35,36,37,38,39,40,41,42,43,44]-CO₂ hydrate, studies on CO₂ solubility in pure water [11,45,46,47,48,49,50,51,52,53] and NaCl aqueous solution [54] in systems without hydrates, as well as studies on CO₂ solubility in pure water [11,12,36,47,48,50,52,53,55,56,57,58,59,60,61] and NaCl aqueous solution [11,36,57,62] in systems with hydrates.

Due to the difficulties in measurements, there are some discrepancies in the reported experimental data, especially CO₂ solubility data, posing challenges to the accuracy of related research. Thus, there is an urgent need for the development of simulation and prediction methods that can be applied to systems encompassing liquid and hydrate phases. Table 1 presents the presently available simulation studies for forecasting CO₂ solubility in pure water/NaCl aqueous solution. Nonetheless, these studies cannot fully meet the research needs of CCUS. For example, some prediction methods, although generally applicable to a wide range of temperature and pressure, did not consider the influence of CO₂ hydrate presence, and some early studies have a limited prediction pressure range for low temperature conditions (below 298.15 K) due to the lack of experimental data support. The current studies have not yet fully explored all conditions related to the CO₂–pure water/NaCl aqueous solution mixtures involved in CCUS technology, nor have they comprehensively addressed the vapor, liquid, and solid hydrate phases.

Table 1. Simulation studies for predicting CO₂ solubility in pure water/NaCl aqueous solution.

Ref.	T/K	P/MPa	System
[45]	274.14~351.31	0.1~12	Pure water
[46]	278.2~318.2	0.1~8	Pure water
[51]	298.15~448.15	0.1~18	Pure water
[53]	274~303	1~4	Pure water
[63]	285.15~373.15	0.1~60	Pure water
[64]	271.65~373.15	0.1~100	Pure water
[65]	271.65~373.15	0.1~100	NaCl aqueous solution
[66]	-	-	NaCl aqueous solution
[67]	273.15~723.15	0.1~150	NaCl aqueous solution
[68]	285.37~373.15	0.1~68.948	Pure water/NaCl aqueous solution
[69,70]	285.15~573.15	0.1~60	Pure water/NaCl aqueous solution
[71]	273~533	0.1~200	Pure water/NaCl aqueous solution
[54]	363	0.1~13	Pure water/NaCl aqueous solution
[36,72,73,74]	Above the critical temperature of CO2 and below 283.15 K	0~50	Pure water/NaCl aqueous solution
[75]	274~300	0.1~100	Pure water/Sea water
[76]	273~533	0.1~200	Pure water/Aqueous solution containing Na+, K+, Ca2+, Mg2+, Cl−, and SO42−
[77]	298~423	0.1~40	Pure water/NaCl, KCl, MgCl2, CaCl2, NaNO3, KNO3, Mg(NO3)2, and Na2CO3 aqueous solution
[78]	273.15~473.65	0.092~71.231	Pure water/NaCl, KCl, MgCl2, CaCl2, MgSO4, K2SO4, NaHCO3, Na2SO4, and mixed salt aqueous solution

In order to overcome the mentioned challenges, this study modified and combined the BWRS EOS (Benedict–Webb–Rubin–Starling equation of state) with hydrate thermodynamic theories for the prediction of CO₂ solubility in pure water/NaCl aqueous solution systems with and without hydrates, under the temperature and pressure range of 273.15~298.15 K and 0.1~40 MPa, as well as the phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate. The discrepancy between the reported experimental results and the calculation results were compared to assess the effectiveness of the methods proposed. Furthermore, CO₂ solubility characteristics and the variation of phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate were described. This study is beneficial for the development of CCUS related technologies.

2. Theory and Methods

Note: The complex calculation processes involving iterative procedures in this study have been organized into flowcharts to enhance the readability (presented in Appendix B to avoid an overly lengthy main text). The flowcharts comprise the following: Figure A1 (for the calculation of density of i-rich phase and vapor pressure of component i, corresponding to Section 2.1.1 and Section 2.3.1), Figure A2 (for the calculation of CO₂ solubility in systems without hydrates, corresponding to Section 2.3.2), Figure A3 (for the calculation of CO₂ solubility in systems with hydrates, corresponding to Section 2.5.1), Figure A4 (for the calculation of dΔμ, corresponding to Section 2.5.2), Figure A5 (for the calculation of phase equilibrium temperature of CO₂–pure water or NaCl aqueous solution–CO₂ hydrate, corresponding to Section 2.5.3), Figure A6 (for the calculation of $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ in systems without hydrates, corresponding to Section 2.6), Figure A7 (for the calculation of $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ in systems with hydrates, corresponding to Section 2.6).

2.1. BWRS EOS

EOSs commonly used to predict the thermodynamic properties of CO₂-related real fluids including Cubic-type EOSs (PR, RK, SRK, GCEOS, etc.), Virial-type EOSs (LKP, BWRS, etc.), and SAFT-type EOSs (PC-SAFT, tPC-SAFT, SAFT-LJ, HRX-SAFT, SAFT-RPM, SAFT-VR, etc.) [79,80,81]. Among them, Cubic-type EOSs are widely used due to their simplicity in form [80]. Virial-type EOSs attract attention for their rigorous theoretical basis [82]. SAFT-type EOSs are renowned for their calculations of polymer mixtures and, to a lesser extent, aqueous systems [83]. However, most EOSs cannot accurately predict the thermodynamic properties of liquid fluids, and their prediction accuracy tends to decrease as the CO₂ content increases [80]. BWRS EOS, the most representative Virial-type EOSs [80], stands out as the first and most well-known EOS that can be applied to both vaporous and liquid fluids [84,85]. It performs highly for specific mixtures, such as synthetic gases, CO, CO₂, H₂, H₂O, etc. [86,87,88]. Therefore, this study selected the BWRS EOS as the fundamental equation for calculating fluid properties.

The basic form of the BWRS EOS [84] is as follows:

$$\begin{array}{r}P=\rho RT+\left(B_{0}RT-A_0-\frac{C_0}{T^2}+\frac{D_0}{T^3}-\frac{E_0}{T^4}\right){\rho }^2+\left(bRT-a-\frac{d}{T}\right){\rho }^3\\ +\alpha \left(a+\frac{d}{T}\right){\rho }^6+\frac{c{\rho }^3}{T^2}\left(1+\gamma {\rho }^2\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\gamma {\rho }^2\right)\hspace{1em}\hspace{1em}\end{array}$$

(1)

where $P$ represents system pressure, MPa; $T$ represents system temperature, K; $\rho$ represents the molar density of the fluid, mol∙mL⁻¹; R is the ideal gas constant, which is 8.314 J∙mol⁻¹∙K⁻¹; and $B_0$, $A_0$, $C_0$, $\gamma$, b, a, $\alpha$, c, $D_0$, d, $E_0$ are 11 parameters related to the fluid itself.

If the fluid consists of a single component, the parameters $B_{0i}$, $A_{0i}$, $C_{0i}$, ${\gamma }_i$, $b_i$, $a_i$, ${\alpha }_i$, $c_i$, $D_{0i}$, $d_i$, and $E_{0i}$ for component i can be calculated based on its critical density (${\rho }_{ci}$), critical temperature ($T_{ci}$), acentric factor (${\omega }_i$) (listed in Table 2), and the universal parameters $A_j$ and $B_j$ (listed in Table 3) as follows:

$${\rho }_{ci}\cdot B_{0i}=A_1+B_1{\omega }_i$$

(2)

$$\frac{{\rho }_{ci}{A}_{0i}}{{RT}_{ci}}=A_2+B_2{\omega }_i$$

(3)

$$\frac{{\rho }_{ci}{C}_{0i}}{R{T}_{ci}^3}=A_3+B_3{\omega }_i$$

(4)

$${\rho }_{ci}^2{\gamma }_i=A_4+B_4{\omega }_i$$

(5)

$${\rho }_{ci}^2{b}_i=A_5+B_5{\omega }_i$$

(6)

$$\frac{{\rho }_{ci}^2{a}_i}{{RT}_{ci}}=A_6+B_6{\omega }_i$$

(7)

$${\rho }_{ci}^3{\alpha }_i=A_7+B_7{\omega }_i$$

(8)

$$\frac{{\rho }_{ci}^2{c}_i}{{RT}_{ci}^3}=A_8+B_8{\omega }_i$$

(9)

$$\frac{{\rho }_{ci}{D}_{0i}}{{RT}_{ci}^4}=A_9+B_9{\omega }_i$$

(10)

$$\frac{{\rho }_{ci}^2{d}_i}{{RT}_{ci}^2}=A_{10}+B_{10}{\omega }_i$$

(11)

$$\frac{{\rho }_{ci}{E}_{0i}}{{RT}_{ci}^5}=A_{11}+B_{11}{\omega }_i\exp (-3.8{\omega }_i)$$

(12)

Table 2. Critical parameters and acentric factors of CO₂ and H₂O [84].

Critical Parameters and Acentric Factors	Component i
	$\mathbf{C}{\mathbf{O}}_2$	${\mathbf{H}}_2\mathbf{O}$
${\rho }_{ci}/\mathrm{m}\mathrm{o}\mathrm{l}·{\mathrm{L}}^{-1}$	10.638	17.857
$T_{ci}/\mathrm{K}$	304.09	647.3
${\omega }_i$	0.21	0.344
$P_{ci}/\mathrm{M}\mathrm{P}\mathrm{a}$	7.376	22.048

Table 3. The values of universal parameters $A_j$ and $B_j$ [84].

Subscript j	Universal Parameters
	${\mathit{A}}_{\mathit{j}}$	${\mathit{B}}_{\mathit{j}}$
1	0.4436900	0.115 449
2	1.2843800	−0.920731
3	0.3563060	1.708710
4	0.5449790	−0.270896
5	0.5286290	0.349621
6	0.4840110	0.754130
7	0.0705233	−0.044448
8	0.5040870	1.322450
9	0.0307452	0.179433
10	0.0732828	0.463492
11	0.0064500	−0.022143

If the fluid consists of multiple components, the 11 parameters in the BWRS EOS can be calculated based on the 11C parameters of C individual components and the proportion ($q_i$) of each component in a particular phase of the mixed fluid using mixing rules [88] as follows:

$$B_0=\sum _{\mathrm{i}=1}^C{q}_i{B}_{0i}$$

(13)

$$A_0={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=1}^c{q}_i{q}_k{A}_{0i}^{\frac{1}{2}}{A}_{0k}^{\frac{1}{2}}\left(1-K_{ik}\right)$$

(14)

$$C_0={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=1}^c{q}_i{q}_k{C}_{0i}^{\frac{1}{2}}{C}_{0k}^{\frac{1}{2}}{\left(1-K_{ik}\right)}^3$$

(15)

$${\gamma }_0={\left(\sum _{i=1}^c{q}_i{\gamma }_i^{\frac{1}{2}}\right)}^2$$

(16)

$$b={\left(\sum _{i=1}^c{q}_i{b}_i^{\frac{1}{3}}\right)}^3$$

(17)

$$a={\left(\sum _{i=1}^c{q}_i{a}_i^{\frac{1}{3}}\right)}^3$$

(18)

$$\alpha ={\left(\sum _{i=1}^c{q}_i{\alpha }_i^{\frac{1}{3}}\right)}^3$$

(19)

$$C={\left(\sum _{i=1}^c{q}_i{C}_i^{\frac{1}{3}}\right)}^3$$

(20)

$$D_0={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=1}^c{q}_i{q}_k{D}_{0i}^{\frac{1}{2}}{D}_{0k}^{\frac{1}{2}}{\left(1-K_{ik}\right)}^4$$

(21)

$$d={\left(\sum _{i=1}^c{q}_i{d}_i^{\frac{1}{3}}\right)}^3$$

(22)

$$E_0={\displaystyle \sum }_{i=1}^c{\displaystyle \sum }_{k=1}^c{q}_i{q}_k{E}_{0i}^{\frac{1}{2}}{E}_{0k}^{\frac{1}{2}}{\left(1-K_{ik}\right)}^5$$

(23)

where the scripts i and k represent components i and k, and $K_{ik}$ represents the interaction coefficient between component i and k. How to acquire $K_{ik}$ through experimental data will be discussed in Section 2.6.

2.1.1. Calculation of Fluid Density

When $P$, $T$, and $q_i$ are all determined, using the Newton iteration method helps achieve rapid convergence in calculating fluid density from BWRS EOS [89]. The BWRS EOS can be reformulated as a function of density [90]:

$$\begin{array}{cc}\hfill F\left(\rho \right)=\rho RT& + \left(B_{0}RT-A_0-\frac{C_0}{T^2}+\frac{D_0}{T^3}-\frac{E_0}{T^4}\right){\rho }^2\hfill \\ & \hspace{1em}+\left(bRT-a-\frac{d}{T}\right){\rho }^3+\alpha \left(a+\frac{d}{T}\right){\rho }^6\hfill \\ & \hspace{1em}+\frac{c{\rho }^3}{T^2}\left(1+\gamma {\rho }^2\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\gamma {\rho }^2\right)-P\hfill \end{array}$$

(24)

A reasonable density root makes $F\left(\rho \right)$ equal to zero. The derivative form of expression (24) is given as follows:

$$\begin{array}{cc}\hfill {\mathrm{F}}^{\prime }\left(\rho \right)=RT& + 2\rho \left(B_{0}RT-A_0-\frac{C_0}{T^2}+\frac{D_0}{T^3}-\frac{E_0}{T^4}\right)\hfill \\ & \hspace{1em}\hspace{1em}+3\left(bRT-a-\frac{d}{T}\right){\rho }^2+6\alpha \left(a+\frac{d}{T}\right){\rho }^5\hfill \\ & \hspace{1em}\hspace{1em}-\frac{2c{{\gamma }^2\rho }^6}{T^2}\exp \left(-\gamma {\rho }^2\right)\hfill \\ & \hspace{1em}\hspace{1em}+\frac{3c{\rho }^2}{T^2}\left(1+\gamma {\rho }^2\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(-\gamma {\rho }^2\right)\hfill \end{array}$$

(25)

The iterative equation for density roots is:

$${\rho }_{n+1}={\rho }_n-\frac{\mathrm{F}\left(\rho \right)}{{\mathrm{F}}^{\prime }\left(\rho \right)}$$

(26)

An initial density value is required for the iteration process. In some cases, Equation (24) may have multiple real roots, and the choice of the initial density value impacts the iteration results. Using the density of ideal fluids as the initial density value for vaporous fluids:

$${\rho }_{0V}=\frac{P}{RT}$$

(27)

Using the weighted result of the Rackett equation [91] as the initial density value for the liquid fluids:

$${\rho }_{0L}=\sum _{i=1}^c\frac{q_i{\rho }_{ci}}{{\left(0.29056-0.08775{\omega }_i\right)}^{\left|{\left(1-\frac{T}{T_{ci}}\right)}^{\frac{2}{7}}\right|}}$$

(28)

Assume the phase state of the fluid before calculating its density. In this study, for the CO₂-rich phase and the H₂O-rich phase, the vapor pressures of the predominant components were used as criteria in the phase determination process. For example, if the system pressure is greater than the vapor pressure of pure CO₂, it is assumed that the CO₂-rich phase is in a liquid state, and ${\rho }_{0L}$ is taken as the initial value for the density calculation. Otherwise, ${\rho }_{0V}$ is taken as the initial value for the density calculation. The calculation methods of vapor pressure of pure CO₂ and H₂O will be introduced in Section 2.3.1.

2.1.2. Calculation of Fugacity Coefficients and Fugacity

The fugacity coefficient φ can be expressed as [84]:

$$\begin{array}{cc}\hfill \ln \phi =Z-1& - \ln Z+\frac{1}{RT}{\int }_0^{\rho }\left(P-\rho RT\right)\frac{d\rho }{{\rho }^2}\hfill \\ & \hspace{1em}=-\ln (\frac{P}{\rho RT})\hfill \\ & \hspace{1em}+\frac{2}{R}\left(B_{0}R-\frac{A_0}{T}-\frac{c_0}{T^3}+\frac{D_0}{T^4}-\frac{E_0}{T^5}\right)\rho \hfill \\ & \hspace{1em}+\frac{3}{2R}\left(bR-\frac{a}{T}-\frac{d}{T^2}\right){\rho }^2+\frac{6\alpha }{5RT}\left(a+\frac{d}{T}\right){\rho }^5\hfill \\ & \hspace{1em}+\frac{c}{\gamma R{T}^3}\left[1+\left(\frac{\gamma {\rho }^2}{2}+{\gamma }^2{\rho }^4-1\right)\exp \left(-\gamma {\rho }^2\right)\right]\hfill \end{array}$$

(29)

where $Z$ represents the compressibility factor, which is defined as:

$$Z=\frac{P}{\rho RT}$$

(30)

The fugacity ($f$) of a single-component fluid satisfies:

$$f=\phi P$$

(31)

For a fluid consisting of multiple components, the fugacity coefficient (${\phi }_i$) of component i can be expressed as [92]:

$$\begin{array}{cc}\hfill \mathrm{l}\mathrm{n}{\phi }_i=\frac{1}{RT}& {\int }_0^{\rho }\left[{\left(\frac{\partial P}{\partial q_i}\right)}_{T,\rho ,q_{k(k\ne i)}}-\rho RT\right]\frac{d\rho }{{\rho }^2}-\ln Z\hfill \\ & \hspace{1em}\hspace{1em}=-\ln (\frac{P}{\rho RT})+\rho \left(B_0+B_{0i}\right)\hfill \\ & \hspace{1em}\hspace{1em}+\frac{1}{RT}\left\{2\rho {\displaystyle \sum _{k=1}^C}{q}_k\right[{-\left(A_{0k}{A}_{0i}\right)}^{\frac{1}{2}}\left(1-K_{ik}\right)\hfill \\ & \hspace{1em}\hspace{1em}-\frac{{\left(C_{0k}{C}_{0i}\right)}^{\frac{1}{2}}}{T^2}{\left(1-K_{ik}\right)}^3+\frac{{\left(D_{0k}{D}_{0i}\right)}^{\frac{1}{2}}}{T^3}{\left(1-K_{ik}\right)}^4\hfill \\ & \hspace{1em}\hspace{1em}-\frac{{\left(E_{0k}{E}_{0i}\right)}^{\frac{1}{2}}}{T_4}{\left(1-K_{ik}\right)}^5]\hfill \\ & \hspace{1em}\hspace{1em}+\frac{{3\rho }^2}{2}\left[{\left(b^2{b}_i\right)}^{\frac{1}{3}}RT-{\left(a^2{a}_i\right)}^{\frac{1}{3}}-\frac{{\left(d^2{d}_i\right)}^{\frac{1}{3}}}{T}\right]\hfill \\ & \hspace{1em}\hspace{1em}+\frac{3\alpha {\rho }^5}{5}\left[{\left(a^2{a}_i\right)}^{\frac{1}{3}}+\frac{{\left(d^2{d}_i\right)}^{\frac{1}{3}}}{T}\right]\hfill \\ & \hspace{1em}\hspace{1em}+\frac{3{\rho }^5}{5}\left(a+\frac{d}{T}\right){\left({\alpha }^2{\alpha }_i\right)}^{\frac{1}{3}}\hfill \\ & \hspace{1em}\hspace{1em}+\frac{{3\left(C^2{C}_i\right)}^{\frac{1}{3}}{\rho }^2}{T^2}[\frac{1-\mathrm{e}\mathrm{x}\mathrm{p}(-\gamma {\rho }^2)}{{\gamma \rho }^2}\hfill \\ & \hspace{1em}\hspace{1em}-\frac{1}{2}\exp (-\gamma {\rho }^2)]\hfill \\ & \hspace{1em}\hspace{1em}-\frac{2C}{{\gamma T}^2}{\left(\frac{{\gamma }_i}{\gamma }\right)}^{\frac{1}{2}}[1-(1+\gamma {\rho }^2\hfill \\ & \hspace{1em}\hspace{1em}+\frac{1}{2}{\gamma }^2{\rho }^4)\exp (-\gamma {\rho }^2\left)\right]\}\hfill \end{array}$$

(32)

The fugacity ($f_i$) of component i satisfies [93]:

$${lnf}_i={ln\phi }_i+\mathrm{l}\mathrm{n}P+\mathrm{l}\mathrm{n}{q}_i$$

(33)

2.2. Principle of Phase Equilibrium

In an equilibrium system, the fugacity of each component in all phases must be equal.

Take superscripts “V”, “L”, “H₂O-rich” and “CO₂-rich” to denote vapor phase, liquid phase, H₂O-rich phase, and CO₂-rich phase, respectively.

For component i in a vapor-liquid equilibrium system:

$$f_i^V=f_i^L$$

(34)

For components H₂O and CO₂ in an equilibrium system of CO₂-pure water/NaCl aqueous solution mixture without hydrates:

$$f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}=f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$$

(35)

$$f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}{=f}_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$$

(36)

For components H₂O and CO₂ in an equilibrium system of pure water/NaCl aqueous solution-CO₂ hydrate:

$$f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}{=f}_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$$

(37)

$$f_{{\mathrm{C}\mathrm{O}}_2}^{\mathrm{H}}{=f}_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$$

(38)

For components H₂O and CO₂ in an equilibrium system of CO₂-CO₂ hydrate:

$$f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}=f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$$

(39)

$$f_{{\mathrm{C}\mathrm{O}}_2}^{\mathrm{H}}{=f}_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$$

(40)

In the equations above, the fugacity in the fluid phases, including vapor and liquid phase in a single-component system, as well as the CO₂-rich and H₂O-rich phase in multi-component systems, can all be calculated using the BWRS EOS.

2.3. Thermodynamic Properties of Systems without Hydrates

2.3.1. Calculation of Vapor Pressure of Component i

The calculation methods of the vapor pressure of component i, $P_i^{*}$, is as follows.

Firstly, an initial value for iteration can be calculable by using the following equation:

$$P_{i_0}^{\mathrm{*}}=P_{ci}·{P_{ci}}^{-1}=P_{ci}·{10}^{\frac{7\left(1+{\omega }_i\right)\left(1-\frac{T_{ci}}{T}\right)}{3}}$$

(41)

where $P_{ci}$ represents the critical pressure, MPa, and its value is listed along with ${\omega }_i$ and $T_{ci}$ in Table 2. The relation between $P_{ci}$ and ${\omega }_i$, $T_{ci}$, and $T$, is provided by Edmister [94].

Then, use $P_{i0}^{*}$ to calculate the fugacity of component i in vapor phase $f_i^V$ and liquid phase $f_i^L$ until Equation (34) is satisfied. If it is not satisfied, the following iterative equation is used to calculate a new vapor pressure until Equation (34) is satisfied:

$$P_{i_{n+1}}^{\mathrm{*}}=P_{i_n}^{\mathrm{*}}\left(1-\frac{\mathrm{l}\mathrm{n}{f}_i^V-\mathrm{l}\mathrm{n}{f}_i^L}{Z_i^V-Z_i^L}\right)$$

(42)

where $Z_i^V$ and $Z_i^L$ represent the compressibility factors of the vapor and liquid phases, respectively.

2.3.2. CO₂ Solubility in Pure Water/NaCl Aqueous Solution

The equilibrium constant of component i between the CO₂-rich phase and H₂O-rich phase, K_i, is giving by its general expression:

$$K_i=\frac{y_i}{x_i}=\frac{\frac{f_i^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}{x_i}}{\frac{f_i^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}{y_i}}=\frac{{\phi }_i^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}{{\phi }_i^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}$$

(43)

where $y_i$ and $x_i$ represent the mole fractions of component i in the CO₂-rich phase and H₂O-rich phase, respectively.

The relationship between the total mole fraction of component i in the fluid phase, $z_i$, and the molar phase fraction, N, is given by:

$$z_i=N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}{y}_i+N_{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}{x}_i$$

(44)

$y_i$ and $x_i$ can be calculated using $K_i$, $z_i$, and $N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ as follows:

$$y_i=\frac{{K_{i}z}_i}{1+N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\left(K_i-1\right)}$$

(45)

$$x_i=\frac{z_i}{1+N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\left(K_i-1\right)}$$

(46)

Subtracting the above two equations gives the Rachford–Rice equation [95]:

$$\sum _i^c\left(y_i-x_i\right)=\sum _i^c\frac{z_i\left(K_i-1\right)}{1+N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\left(K_i-1\right)}=0$$

(47)

To use the Newton iteration method to calculate $N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$, define $F\left(N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\right)$:

$$F\left(N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\right)=\sum _i^c\frac{z_i\left(K_i-1\right)}{1+N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\left(K_i-1\right)}$$

(48)

A reasonable $N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ value makes $F\left(N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\right)$ equal to zero. An initial value for $N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ is chosen as:

$${N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}_0=0.5\sum _i^c\frac{1}{1-K_i}$$

(49)

The derivative form of expression (48) is:

$$F^{\prime }\left(N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\right)=-\sum _i^c\frac{{\mathrm{z}}_i{\left(K_i-1\right)}^2}{{\left[1+N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\left(K_i-1\right)\right]}^2}$$

(50)

The iterative equation for $N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ is:

$${N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}_{n+1}={N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}_n-\frac{F\left(N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\right)}{F^{\prime }\left(N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}\right)}$$

(51)

The steps to predict the mole fractions of component i in the CO₂-rich phase and H₂O-rich phase, $y_i$ and $x_i$, are as follows:

(1)

Assume initial values for K_i; recommended values are 62 and 0.001 for CO₂ and for H₂O, respectively.

(2)

Calculate $N_{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ using Equations (48)–(51).

(3)

Calculate $y_i$ and $x_i$ using Equations (45) and (46).

(4)

Use the method described in Section 2.4.2 to calculate $f_i^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ and $f_i^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ until Equations (35) and (36) are satisfied. If they are not satisfied, modify $y_i$ and $x_i$ and repeat steps (2) to (3) until they are satisfied. Based on multiple actual simulation results, the recommended method for modifying $K_i$ is:

$${K_i}_{n+1}={K_i}_n\frac{f_i^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}{f_i^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}}$$

(52)

CO₂ solubility in pure water/NaCl aqueous solution, $S_{{\mathrm{C}\mathrm{O}}_2}$, is:

$$S_{{\mathrm{C}\mathrm{O}}_2}=\frac{x_{{\mathrm{C}\mathrm{O}}_2}}{x_{{\mathrm{H}}_2\mathrm{O}}}$$

(53)

2.4. Hydrate Thermodynamic Theories

CO₂ hydrate is a type I hydrate, containing 46 H₂O molecules [29]. The H₂O molecules form 2 small cavities and 6 large cavities that can store guest molecules (normally not all cavitied are filled with CO₂) [96]. According to the fundamental equation of chemical potential, the fugacity of a certain component in system-A can be calculated by the fugacity of this component in system-B and the chemical potential difference of this component between these two systems. The hydrate thermodynamic theories provide the chemical potential difference of component H₂O in the empty hydrate cavities (without guest molecules) and a specific system, which can be used to calculate the fugacity of H₂O in a specific system.

2.4.1. Van der Waals–Platteeuw Model

The Van der Waals–Platteeuw (VdWP) model [96] gives the chemical potential difference of H₂O in empty hydrate cavities and in hydrate cavities that contain guest molecule k ($\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$) as follows:

$$\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}=-RT\sum _{j=1}^2{v}_j\mathrm{l}\mathrm{n}\left(1-\sum _{k=1}^c{\theta }_{kj}\right)$$

(54)

$${\theta }_{kj}=\frac{f_k{C}_j}{1+f_k{C}_j}(j=S,L)$$

(55)

where “S” denotes small cavities and “L” denotes large cavities. $v_j$ represents the moles of j-type hydrate cavities formed by 1 mole of H₂O molecules. For CO₂ hydrate and other type I hydrates, the moles of small cavities is 1/23 and the moles of large cavities is 3/23 [96]. ${\theta }_{kj}$ represents the occupancy of component k in j-type hydrate cavities. $f_k$ represents the fugacity of the guest molecule k. For equilibrium system containing CO₂ hydrate, $f_k$ refers to $f_{{\mathrm{C}\mathrm{O}}_2}^{\mathrm{H}}$, $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$, or $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$; the last two can be calculated using BWRS EOS. $C_j$ is the Langmuir constant, and in this study, it was calculated using three methods that will be introduced in Section 2.4.2.

For CO₂ hydrate and other type I hydrates, the expression for $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ can be organized as:

$$\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}=\frac{RT}{23}\mathrm{l}\mathrm{n}\left[\left(1+f_k{C}_S\right){\left(1+f_k{C}_L\right)}^3\right]$$

(56)

2.4.2. Calculation of Langmuir Constants

$C_j$ can be obtained through experimental data fitting or derived from thermodynamic theory. However, due to variations in experimental conditions and theoretical assumptions, different studies may yield different values of $C_j$. In order to ascertain the most suitable value of $C_j$ for the research systems of the present study, three classical research methods for calculating $C_j$ were compared.

The first method refers to [97,98], and the calculation is as follows:

$$C_S=\frac{1.6464}{100000}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{2799.66}{T-15.9}\right)$$

(57)

$$C_L=\frac{0.8507}{0.101325\mathrm{T}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{3277.9}{T}\right)$$

(58)

The second method refers to [99], and the calculation is as follows:

$$C_S=\frac{0.0000566}{T}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{4182.52}{T}+\frac{44770}{T^2}\right)$$

(59)

$$C_L=\frac{0.007879}{T}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{0.00364536}{T}+\frac{31390}{T^2}\right)$$

(60)

The third method refers to [98,99], and the calculation is as follows:

$$C_S=\frac{0.0000566}{T}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{4182.52}{T}+\frac{44770}{T^2}\right)$$

(61)

$$C_L=\frac{0.8507}{0.101325\mathrm{T}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{3277.9}{T}\right)$$

(62)

2.4.3. Saito Model and Its Modification

Saito et al. [100] provided a model to calculate the chemical potential difference of H₂O in empty hydrate cavities and in pure water ($\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$). Holder et al. [101] further organized and improved the model, which is expressed as:

$$\frac{\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}}{RT}=\frac{\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^0}{RT}-{\int }_{T_0}^T\left(\frac{\mathsf{\Delta }{h}_{{\mathrm{H}}_2\mathrm{O}}}{{RT}^2}\right)dT+{\int }_{P_0}^P\left(\frac{\mathsf{\Delta }{V}_{{\mathrm{H}}_2\mathrm{O}}}{RT}\right)dP$$

(63)

$$\mathsf{\Delta }{h}_{{\mathrm{H}}_2\mathrm{O}}=\mathsf{\Delta }{h}_{{\mathrm{H}}_2\mathrm{O}}^0+{\int }_{T_0}^T\mathsf{\Delta }{C}_{P_{{\mathrm{H}}_2\mathrm{O}}}dT$$

(64)

$$\mathsf{\Delta }{C}_{P_{{\mathrm{H}}_2\mathrm{O}}}=\mathsf{\Delta }{C}_{P_{{\mathrm{H}}_2\mathrm{O}}}^0-{\int }_{T_0}^{T}0.141dT$$

(65)

where the superscript 0 represents the reference state (temperature of 273.15 K and pressure of 0 MPa). $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^0$ represents the chemical potential difference of H₂O in empty hydrate cavities and pure water at the reference state, and $\mathsf{\Delta }{h}_{{\mathrm{H}}_2\mathrm{O}}^0$ and $\mathsf{\Delta }{C}_{P_{{\mathrm{H}}_2\mathrm{O}}}^0$ represent the enthalpy and molar heat capacity difference of H₂O in hydrate phase and pure water at the reference state. For type I hydrates, their values are taken as 1264.13 J∙mol⁻¹, −4861.03 J∙mol⁻¹, and −38.13 J∙mol⁻¹∙K⁻¹, respectively [98]. $\mathsf{\Delta }{h}_{{\mathrm{H}}_2\mathrm{O}}$, $\mathsf{\Delta }{V}_{{\mathrm{H}}_2\mathrm{O}}$, and $\mathsf{\Delta }{C}_{P_{{\mathrm{H}}_2\mathrm{O}}}$ represent the enthalpy, volume, and molar heat capacity difference of empty hydrate cavities and pure water in any state.

$\mathsf{\Delta }{V}_{{\mathrm{H}}_2\mathrm{O}}$ is often considered a constant value [101,102]. This is a result of assuming that the difference between the density of empty hydrate cavities (${\rho }_{\mathrm{H}}^{\mathrm{E}\mathrm{H}},\mathrm{g}·{\mathrm{m}\mathrm{L}}^{-1}$) and pure water (${\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$) do not vary with temperature and pressure, which makes $\mathsf{\Delta }{V}_{{\mathrm{H}}_2\mathrm{O}}$ independent of guest molecules and theoretically can be used for the calculation of all Type I hydrates. However, the actual ${\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ varies with temperature and pressure conditions, and recent research results [103] have shown that ${\rho }_{\mathrm{H}}^{\mathrm{E}\mathrm{H}}$ relates to the type of guest molecules, as well as temperature and pressure conditions. Therefore, to expand the applicable temperature and pressure range of the above model and improve its accuracy, this study did not consider $\mathsf{\Delta }{V}_{{\mathrm{H}}_2\mathrm{O}}$ to be constant, but calculated it according to its original definition as follows:

$$\mathsf{\Delta }{V}_{{\mathrm{H}}_2\mathrm{O}}=\frac{M_{{\mathrm{H}}_2\mathrm{O}}}{{\rho }_{\mathrm{H}}^{\mathrm{E}\mathrm{H}}}-\frac{1}{{\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}}$$

(66)

where $M_{{\mathrm{H}}_2\mathrm{O}}$ represents the relative molecular mass of H₂O, which is taken as 18 g∙mol⁻¹. ${\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ is calculated using the BWRS EOS. A set of data for ${\rho }_{\mathrm{H}}^{\mathrm{E}\mathrm{H}}$ under different temperature and pressure conditions is provided in reference [103]. Based on these data, an equation is fitted to calculate ${\rho }_{\mathrm{H}}^{\mathrm{E}\mathrm{H}}$, with a goodness of fit of 0.99452 and a residual sum of squares of 0.00101302.

$$\mathsf{\Delta }{\rho }_{\mathrm{H}}^{EH}=0.97477609-0.00071519T+0.00086046\mathrm{l}\mathrm{n}P$$

(67)

The expression for $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$ can be organized as:

$$\begin{array}{cc}\hfill \mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}=T[\frac{∆{\mu }_{{\mathrm{H}}_2\mathrm{O}}^0}{T_0}& +\mathsf{\Delta }{h}_{{\mathrm{H}}_2\mathrm{O}}^0\left(\frac{1}{T}-\frac{1}{T_0}\right)\hfill \\ & +\mathsf{\Delta }{C}_{P_{{\mathrm{H}}_2\mathrm{O}}}^0\left(\ln \frac{T_0}{T}-\frac{T_0}{T}+1\right)\hfill \\ & +0.0705\left(\frac{T_0^2}{T}-T\right)+0.141T_0\ln \frac{T}{T_0}]\hfill \\ & +\left(\frac{M_{{\mathrm{H}}_2\mathrm{O}}}{{\rho }_H^{EH}}-\frac{1}{{\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}}\right)\left(P-P_0\right)\hfill \end{array}$$

(68)

2.4.4. Calculation of H₂O Fugacity in CO₂ Hydrate Phase

Based on the thermodynamic theory mentioned above, the fugacity of H₂O in CO₂ hydrate phase, $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$, can be expressed as:

$$f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}=f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{E}\mathrm{H}}{\mathrm{e}}^{-\frac{\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}}{RT}}$$

(69)

In the equation, $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{E}\mathrm{H}}$ represents the fugacity of H₂O in empty hydrate phase:

$$f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{E}\mathrm{H}}=f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}{\mathrm{e}}^{\left(\frac{\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}}{RT}\right)}$$

(70)

where $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ represents the fugacity of H₂O in pure water.

The expression for $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ can be rearranged as follows:

$$f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{E}\mathrm{H}}=f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}{e}^{\left(\frac{\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}}{RT}\right)}$$

(71)

2.5. Thermodynamic Properties of Systems with Hydrates

2.5.1. CO₂ Solubility in Pure Water/NaCl Aqueous Solution

Reasonable CO₂ solubility equalizes the two values of $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ calculated by using Equations (37) and (71).

The steps for predicting the mole fractions of component i, $x_i$ in H₂O-rich phase are as follows:

(1)

Use the method described in Section 2.1 to calculate ${\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$.

(2)

Calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$ using Equation (68).

(3)

Assume an initial value for $x_i$, recommended values are 0.03 and 0.97 for CO₂ and for H₂O, respectively.

(4)

Use the method described in Section 2.1.2 to calculate $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$.

(5)

Calculate the first value of $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ using Equation (37) and denote it as $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$.

(6)

Calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ using Equation (56).

(7)

Calculate the second value of $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ using Equation (71) and denote it as $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$.

(8)

Compare $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$; if they are not equal, modify $x_i$ using the following equation and repeat steps (4) to (7) until they are equal:

$${x_{{\mathrm{C}\mathrm{O}}_2}}_{n+1}={x_{{\mathrm{C}\mathrm{O}}_2}}_n\frac{f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}}{f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}}$$

(72)

$${x_{{\mathrm{H}}_2\mathrm{O}}}_{n+1}=1-{x_{{\mathrm{C}\mathrm{O}}_2}}_{n+1}$$

(73)

Calculate the CO₂ solubility using Equation (53).

2.5.2. Modification of the VdWP Model

In this study, BWRS EOS for systems with hydrates was modified with the reported experimental CO₂ solubility data (the modification will be introduced in Section 2.6), which only leads to accurate calculation results for pure water/NaCl aqueous solution–CO₂ hydrate system. In the meantime, due to a lack of sufficient reported experimental data on the composition of the CO₂-rich phase, BWRS EOS could not be modified to suit the calculation for the CO₂-rich phase. Consequently, under the reported experimental phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate, using $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ to calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ will yield a different result than using $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ to calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$. This discrepancy was used to modify the calculation results for the VdWP model, which will expand the application scope of the method proposed in this study to CO₂-CO₂ hydrate systems and CO₂–pure water/NaCl aqueous solution–CO₂ hydrate systems.

Firstly, calculate $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ using the BWRS EOS modified for pure water/NaCl aqueous solution–CO₂ hydrate system, steps are as follows:

(1)

Use the method described in Section 2.1 to calculate ${\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$.

(2)

Calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$ using Equation (68).

(3)

Assume an initial value for $y_i$; the recommended values are 0.99999999 and 0.00000001 for CO₂ and for H₂O, respectively.

(4)

Use the method described in Section 2.1.2 to calculate $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$.

(5)

Calculate the first value of $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ using Equation (39) and denote it as $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$.

(6)

Calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ with $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$, using Equation (56).

(7)

Calculate the second value of $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ using Equation (71) and denote it as $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$.

(8)

Compare $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$; if they are not equal, modify $y_i$ using the following equation and repeat steps (4) to (7) until they are equal:

$${y_{{\mathrm{C}\mathrm{O}}_2}}_{n+1}={y_{{\mathrm{C}\mathrm{O}}_2}}_n-0.00000001$$

(74)

$${y_{{\mathrm{H}}_2\mathrm{O}}}_{n+1}=1-{y_{{\mathrm{C}\mathrm{O}}_2}}_{n+1}$$

(75)

(9)

Use the method described in Section 2.1.2 to calculate $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$.

(10)

Calculate the first $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ with $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ using Equation (56).

And then, use the method described in Section 2.5.1 to predict $x_{{\mathrm{C}\mathrm{O}}_2}$ and $x_{{\mathrm{H}}_2\mathrm{O}}$ at pure water/NaCl aqueous solution–CO₂ hydrate equilibrium, and calculate the second $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ with $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ following steps (4) to (6) in Section 2.5.1. Finally, denote the difference of the two $\Delta {\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ values as $d\mathsf{\Delta }\mu$, and fit $d\mathsf{\Delta }\mu$ as a function of temperature ($T$) and the mass fraction of NaCl in aqueous solution (${\mathrm{S}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l},}, \mathrm{w}\mathrm{t}\%$).

After this modification, when using $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{C}\mathrm{O}}_2-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ to calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ with Equation (56), add $d\mathsf{\Delta }\mu$ from the result for subsequent calculations.

2.5.3. Phase Equilibrium Conditions of CO₂–Pure Water/NaCl Aqueous Solution–CO₂ Hydrate

The steps for predicting the phase equilibrium temperature $T^{eq}$ of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate at a specific pressure P are as follows:

(1)

Assume an initial value for $T^{eq}$, such as 273.15 K.

(2)

Use the method described in Section 2.5.1 to predict $x_{{\mathrm{C}\mathrm{O}}_2}$ and $x_{{\mathrm{H}}_2\mathrm{O}}$, and calculate the first $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ with $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$, following steps (4) to (6) in Section 2.5.1.

(3)

Follow steps (1) to (10) in Section 2.5.2 to calculate a $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ value and add a $d\mathsf{\Delta }\mu$ value to it as the second value of $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$.

(4)

Compare the difference between the two $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ values; if they are not equal, modify $T^{eq}$ with the following equation and repeat steps (2) to (3) until they are equal:

$${T^{eq}}_{n+1}={T^{eq}}_n+0.01$$

(76)

2.6. Modification of BWRS EOS

When calculating the properties of mixed fluids, some of the parameters in the BWRS EOS are influenced by the interaction coefficient between component i and component k, $K_{ik}$ [88]. The $K_{ik}$ value ranges from 0 to 1, and a higher value indicates a greater influence of the mixing effect on the thermodynamic properties of the mixed fluids calculated by the BWRS EOS. When the thermodynamic properties of two fluids are similar and their interaction is weak, the differences in thermodynamic properties between their mixtures and the original fluids will also be smaller. The current reported $K_{ik}$ value is 0 for CO₂ and H₂O [90]. However, with this value, the BWRS EOS is often unsolvable when calculating the thermodynamic properties of CO₂-pure water/NaCl aqueous solution mixtures involving liquid and hydrate phases. This limitation not only hinders comprehension of fluid properties but also restricts the practical application of the BWRS EOS. To overcome this limitation, the reported experimental CO₂ solubility ($S_{{\mathrm{C}\mathrm{O}}_2}^{ref}$) data were used to modify the $K_{ik}$ value ($K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}})$.

For the equilibrium system of CO₂–pure water/NaCl aqueous solution mixtures without hydrates, the modification steps are as follows:

(1)

Assume an initial value for $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$; the recommended value is 0.3.

(2)

Use the method described in Section 2.3.2 to predict the mole fraction of CO₂ and H₂O in the H₂O-rich phase, $x_{{\mathrm{C}\mathrm{O}}_2}$ and $x_{{\mathrm{H}}_2\mathrm{O}}$, at pure water/NaCl aqueous solution equilibrium.

(3)

Calculate CO₂ solubility ($S_{{\mathrm{C}\mathrm{O}}_2}^{cal}$) using Equation (53).

(4)

Compare $S_{{\mathrm{C}\mathrm{O}}_2}^{cal}$ with $S_{{\mathrm{C}\mathrm{O}}_2}^{ref}$; if they are not equal, modify $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ using the following equation and repeat steps (2) and (3) until they are equal:

$${K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}}_{n+1}={K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}}_n-0.00001$$

(77)

For the equilibrium system of CO₂–pure water/NaCl aqueous solution mixtures with hydrates, the modification steps are as follows:

(1)

Use the method described in Section 2.1 to calculate ${\rho }_{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$.

(2)

Calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$ using Equation (68).

(3)

Assume an initial value of 0.2 for $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$.

(4)

Calculate $x_{{\mathrm{C}\mathrm{O}}_2}$ and $x_{{\mathrm{H}}_2\mathrm{O}}$ according to $S_{{\mathrm{C}\mathrm{O}}_2}^{ref}$ as follows:

$$x_{{\mathrm{C}\mathrm{O}}_2}=\frac{S_{{\mathrm{C}\mathrm{O}}_2}^{ref}}{S_{{\mathrm{C}\mathrm{O}}_2}^{ref}+1}$$

(78)

$$x_{{\mathrm{H}}_2\mathrm{O}}=\frac{1}{S_{{\mathrm{C}\mathrm{O}}_2}^{ref}+1}$$

(79)

(5)

Use the method described in Section 2.1.2 to calculate $f_{{\mathrm{C}\mathrm{O}}_2}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$ and $f_{{\mathrm{H}}_2\mathrm{O}}^{{\mathrm{H}}_2\mathrm{O}-\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}}$.

(6)

Use Equation (37) to calculate the first $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ value and denote it as $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$.

(7)

Calculate $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ using Equation (56).

(8)

Use Equation (71) to calculate the second $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H}}$ value and denote it as $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$.

(9)

If $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ is not equal to $f_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},2}$, modify $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ using Equation (77) and repeat steps (5) to (8) until they are equal.

Fit $K_{{CO}_2-H_{2}O}$ as a function of temperature ($T$), pressure ($P$), and the mass fraction of NaCl in aqueous solution (${\mathrm{S}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}$).

2.7. Statistics and Screening of Reported Experimental Data

The reported experimental data, including CO₂ solubility and the phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate, within the temperature range of 273.15~298.15 K and the pressure range of 0~40 MPa, are summarized in Table A1, Table A2 and Table A3. Research systems were distinguished based on whether they contain hydrates and NaCl.

Although this study focuses on the temperature range of 273.15~298.15 K and pressure range of 0.01~40 MPa, if the experimental research in the relevant literature involves both within this range and beyond, the data from the literature were used to modify the BWRS EOS and hydrate thermodynamic model except for some obviously abnormal values, which is to ensure the internal consistency of the prediction methods proposed in this study and expand their extrapolation capability.

The studies of CO₂ solubility in pure water in systems without hydrates [11,33,45,46,47,48,49,50,51,52,53] consists of 361 data points in total. The overall temperature range of the studies is 273.15~573.15 K and the pressure range is 0.01~120 MPa. Among these, the literature [46,47,49,50,52] provides a relatively rich amount of data with 251 data points in total that cover the temperature range of 273.15~573.15 K and the pressure range of 0.01~120 MPa. These data from selected studies were used to modify the BWRS EOS and to evaluate the fitting accuracy. Data from the remaining studies [11,45,48,51,53,104] were used to evaluate the prediction accuracy.

The studies of CO₂ solubility in NaCl aqueous solutions in systems without hydrates [54] consist of 44 data points in total. The overall temperature range of the study is 293.08~353.23 K, the pressure range is 1.02~14.29 MPa, and the NaCl mass fraction range is 1.02~14.29 wt%. All of the data were used to modify the BWRS EOS and to evaluate the fitting accuracy.

The studies of CO₂ solubility in pure water in systems with hydrates [11,12,36,47,48,50,52,53,55,56,57,58,59,61,62] consist of 285 data points in total. The overall temperature range of the studies is 273.15~293.15 K and the pressure range is 1.33~120 MPa. The data in studies [52,55,59] exhibits significant deviations from the data in other studies and were not used to modify BWRS EOS and evaluate the fitting accuracy. Studies [12,56,58,61] provide a relatively rich amount of data with 251 data points in total that covers the temperature range of 273.15~289.05 K and pressure range of 1.874~90 MPa. These data from selected studies were used to modify the BWRS EOS and to evaluate the fitting accuracy. Data from the remaining studies [11,36,48,50,53,57,62] were used to evaluate the prediction accuracy.

The studies of CO₂ solubility in NaCl aqueous solution in systems with hydrates [11,36,57,62] consist of 70 data points in total. The overall temperature range of the studies is 271.45~281.15 K, the pressure range is 1.45~21.79 MPa, and the NaCl mass fraction range is 1~5.844 wt%. All of the data were used to modify the BWRS EOS and to evaluate the fitting accuracy.

The studies of the phase equilibrium conditions of CO₂-pure water-CO₂ hydrate [11,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] consist of 252 data points in total. The overall temperature range of the studies is 269.45~294 K and the pressure range is 0.825~494 MPa. However, there are only 32 data points within the wide pressure range of 40~494 MPa. Considering that this pressure range is not within the scope of this study, 220 data from these studies within the temperature range of 269.45~286.2 K and the pressure range of 0.825~37.2 MPa were selected to modify the BWRS EOS and to evaluate the fitting accuracy.

The studies of the phase equilibrium conditions of the CO₂–NaCl aqueous solution–CO₂ hydrate [11,20,30,31,32,34,35,36,37,38,39,40,41,42,43] consist of 231 data points in total. The overall temperature range of the studies is 254.75~284.6 K, the pressure range is 0.703~60.14 MPa, and the NaCl mass fraction range is 2~25 wt%. All of the data were used to modify the hydrate thermodynamic models and to evaluate the fitting accuracy.

3. Results and Discussion

In this study, modifications were made to the BWRS EOS and VdWP models. At the same time, three different methods were used to calculate the Langmuir constants. For convenience, these methods are denoted as method 1, 2, and 3 in the later text.

3.1. Modification of BWRS EOS

The interaction coefficient between CO₂ and H₂O $(K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$) in the BWRS EOS was fitted and expressed as a function of temperature ($T$), pressure ($P$) and the mass fraction of NaCl in aqueous solution ($S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}$). The fitting equations are shown in Table 4. The calculated $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ based on experimental data are very different from the original $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ value 0 [90], which demonstrate exactly as introduced in detail in 2.5 that this modification is necessary for calculating the thermodynamic properties of CO₂–pure water/NaCl aqueous solution mixtures involving liquid and hydrate phases. This is because as the pressure increases, the difference of thermodynamic properties between the actual fluid and ideal fluid increases. At the same time, the presence of hydrogen bonds between CO₂ and the pure water/NaCl aqueous solution, as well as the hydrogen bonds in the hydrate cavities, exacerbates this difference. As a result, predicting the thermodynamic properties of CO₂-pure water/NaCl aqueous solution mixtures with liquid phase or hydrate phase is difficult using the general EOS. And therefore, in the application of BWRS EOS, the interaction between CO₂ and pure water/NaCl aqueous solution should not be ignored for the system of CO₂–pure water/NaCl aqueous solution mixtures containing a liquid phase and a hydrate phase.

Table 4. Fitting equations of $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ regarding temperature, pressure, and NaCl mass fraction.

System	Method	${\mathit{K}}_{{\mathbf{C}\mathbf{O}}_2-{\mathbf{H}}_2\mathbf{O}}$ Equation
Without hydrate	-	$-0.00000000489913\times T^3+0.00000576416\times T^2-0.00234000\times T-0.00223000\times lnP+0.433930-S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}\times (0.00160000-0.00000805212\times T+0.0000000264483\times P)$
With hydrate	1	$0.406830-0.00102000\times T+0.00000888515\times P-S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}\times (0.023930-0.0000884302\times T+0.0000400857\times P)$
	2	$0.350280-0.000795855\times T+0.00000215094\times P-S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}\times (0.0149300-0.0000576566\times T+0.0000539784\times P)$
	3	$0.398170-0.000998231\times T+0.00000629555\times P-S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}\times (0.0166500-0.0000638813\times T+0.0000545214\times P)$

Theoretically, the closer the thermodynamic properties of the two fluids are, and the smaller the interactions are, the less difference there will be between the thermodynamic properties of their mixtures and the original fluids; correspondingly, the value of $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ will be smaller. At normal temperature and pressure, CO₂ is in a vapor state with relatively large intermolecular distances. In contrast, the pure water/NaCl aqueous solution remains in a liquid state with a lower molecular kinetic energy. The thermodynamic properties of CO₂ and water/NaCl aqueous solution exhibit a significant disparity. As the temperature and pressure increase, the kinetic energy of the H₂O molecules increase and the intermolecular distances of CO₂ decrease, respectively, both effects narrowing the thermodynamic properties gap between vaporous CO₂ and the pure water/NaCl aqueous solution, and therefore, $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ decreases. The $K_{{\mathrm{C}\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}}$ fitting equation for the system without hydrates and NaCl reflects this rule, which is conductive to better reflecting the interaction between CO₂ and pure water/NaCl aqueous solution, thereby improving the accuracy of predicting the thermodynamic properties of their mixtures. However, this rule does not apply to systems containing hydrate. This is because the formation of strong hydrogen bonds within the hydrate cavities reduces the influence of pressure on the size of the hydrate cavities and the distance between the CO₂ and H₂O molecules. Similarly, the rule does not hold true for NaCl-containing systems, as the presence of ions complicates the thermodynamic properties of the system.

3.2. Calculation Results of CO₂ Solubility

Figure 1 displays a scatter plot comparing the reported experimental values of CO₂ solubility (horizontal axis) with the calculated values of this study (vertical axis), illustrating the correlation between the two sets of results. As shown in Figure 1a,b, for a system without hydrates and NaCl, except for a few points, the vast majority of points align closely along the line of y = x. This alignment suggests a strong consistency between the calculated CO₂ solubility values and the experimental values. In Figure 1c,d, it can be observed that the calculation effects of the three calculation methods with different Langmuir constants values (introduced in Section 2.4.2) are nearly identical for both the pure water system and NaCl aqueous solution system with hydrates. While the calculated CO₂ solubility values match most of the experimental values, they significantly deviate from some of the experimental values due to large variations in data from different studies. The calculation methods proposed in this study reconciled the discrepancies among the different studies.

Figure 1. Correlation between the reported experimental values and the calculated values of CO₂ solubility in pure water/NaCl aqueous solution. (a) Pure water system without hydrates; (b) NaCl aqueous solution system without hydrates; (c) pure water system with hydrates; (d) NaCl aqueous solution system with hydrates.

Figure 2 shows the calculated values of CO₂ solubility in different systems and some reported experimental values. The trend of CO₂ solubility calculated in this study is consistent with most of the experimental data. In a pure water system without hydrates (Figure 2a), the calculated trend of CO₂ solubility closely aligns with experimental data, exhibiting an increase with decreasing temperature and increasing pressure. In a pure water system with hydrates (Figure 2b), data from references [56,61] clearly show a decrease in CO₂ solubility as the temperature rises. However, the influence of pressure remains ambiguous, with varying trends reported across different sources, including some showing no clear pattern (reference [11]). To resolve the ongoing controversy regarding the pressure effect, reference [61] measured CO₂ solubility across a wide range of temperatures and pressures (276.15~289.05 K, 3~90 MPa), revealing a slight decrease in solubility with increasing pressure. The findings appear to be reliable and it can be explained that CO₂ is more stably encaged in hydrate cavities at high pressure condition. Additionally, data from reference [52] suggest that in a pure water system with hydrates, CO₂ solubility follows a similar trend to that observed in a system without hydrates, increasing with decreasing temperature and increasing pressure. We speculate that in the experiments reported by reference [52], hydrates may not have formed despite conditions favoring its formation, potentially leading to discrepancies in their measurements compared to other sources. In summary, the trends reported in references [56,61] regarding the variation of CO₂ solubility with temperature and pressure are deemed credible. The calculation results are in alignment with these findings, confirming that CO₂ solubility increases with higher temperatures and lower pressures.

Figure 2. CO₂ solubility in pure water/NaCl aqueous solution. (a) Pure water system without hydrates; (b) pure water system with hydrates; (c) NaCl aqueous solution system without hydrates; (d) NaCl aqueous solution system with hydrates. Hollow circle symbols represent experimental [11,36,45,46,47,48,49,50,52,53,56,60,61] data.

In a NaCl aqueous solution system without hydrates (Figure 2c), CO₂ solubility increases with decreasing temperature, increasing pressure, and decreasing NaCl mass fraction. In a NaCl aqueous solution system with hydrates (Figure 2d), CO₂ solubility increases with increasing temperature. At low pressure, the solubility decreases with increasing NaCl mass fraction, but at high pressure, it increases with increasing NaCl mass fraction. As the NaCl mass fraction increases, the degree of increase in CO₂ solubility with increasing pressure also increases. Under high-pressure conditions, particularly in a system with hydrates (Figure 2b,d), the impact of pressure is found to be lower when compared to a low-pressure system without hydrates (Figure 2a,c). These findings suggest that the existence of hydrates and NaCl in the system alters the solubility characteristics of CO₂.

Table 5 summarizes the fitting and prediction accuracy of the methods proposed in this study for calculating CO₂ solubility in a pure water/NaCl aqueous solution. The fitting accuracy is assessed by comparing the calculated results with the original data used for fitting. The results show that for a pure water system, the fitting accuracy, measured by the mean of the average absolute relative deviation (AARD), ranges from 4.318% to 4.472%. For a NaCl aqueous solution system, the fitting accuracy ranges from 4.318% to 8.472% (AARD). The prediction accuracy is assessed by comparing the calculated results with data that were not involved in the fitting process. The results show that for a pure water system, the prediction accuracy ranges from 7.912 to 17.231% (AARD). Among the three methods used to calculate Langmuir constants, method 1 exhibited the highest prediction accuracy, followed by method 2, and then method 3. But the overall difference is not significant.

Table 5. The calculation effect of CO₂ solubility.

System	Methods	Fitting Data Points	Fitting Accuracy/%	Prediction Data Points	Prediction Error/%
Pure water without hydrates	-	251	5.293	114	7.912
NaCl aqueous solution without hydrates		44	4.318	-	-
Pure water with hydrates	1	119	5.953	118	12.930
	2	119	4.549	118	15.174
	3	119	4.538	118	15.154
NaCl aqueous solution with hydrates	1	70	7.500	-	-
	2	70	8.459	-	-
	3	70	8.472	-	-

The discrepancy statistics across different studies can be found in Table A1 and Table A2. Given that the methods proposed in this study are informed by reported experimental data, they possess a high level of credibility. Consequently, the CO₂ solubility characteristics identified in the literature with smaller AARDs exhibit a degree of consistency, while those identified in the literature with larger AARDs demonstrate discrepancies from other sources.

Before modifications were made to the BWRS EOS, it was not suitable for calculating CO₂ solubility in systems containing a liquid phase and hydrate phase. However, after modifications, using the BWRS EOS combined with hydrate thermodynamic theories can now predict the CO₂ solubility in complex systems. The fitting accuracy and prediction accuracy reached the level of experimental accuracy.

3.3. Modification of VdWP Model

As described in Section 2.5.2, the customized parameter $d\mathsf{\Delta }\mu$ is employed to reconcile the discrepancy between the two $\mathsf{\Delta }{\mu }_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{H},1}$ values calculated using the fugacity of CO₂ in the CO₂-rich phase and in a pure water/NaCl aqueous solution, respectively. The fitting equations of $d\mathsf{\Delta }\mu$ are presented in Table 6, which are derived from fitting the reported experimental data of the phase equilibrium conditions of a CO₂–pure water/NaCl aqueous solution–CO₂ hydrate.

Table 6. Fitting equations of $d\Delta \mu$.

Method	$\mathit{d}\mathsf{\Delta }\mathit{\mu }$ Equation
1	$730-2.8\times T+(0.0422-0.000646061\times S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}-0.0000390636\times T)\times T\times S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}$
2	$1600-6.49\times T+(52.48028+0.42275\times S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}-0.10537\times T)\times S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}$
3	$820-3\times T+(30.37533+0.15524\times S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}-0.005197\times T)\times S_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}$

3.4. Calculation Results of CO₂–Pure Water/NaCl Aqueous Solution–CO₂ Hydrate Phase Equilibrium Conditions

The horizontal axis of Figure 3 displays the reported experimental values for the phase equilibrium temperature of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate at a specific pressure, while the vertical axis shows the values calculated by the methods proposed in this study at the same pressure. For a pure water system, the calculated results closely match the experimental results as depicted in Figure 3a. On the other hand, for an NaCl aqueous solution system, the calculations using method 3 most closely align with the experimental results, while method 1 overestimates and method 2 underestimates the phase equilibrium temperature at a specific pressure, as shown in Figure 3b.

Figure 3. Correlation between the reported experimental values and the calculated values of this study on the phase equilibrium temperature of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate under specific pressure conditions. (a) Pure water system; (b) NaCl aqueous solution system.

Figure 4 illustrates the calculated values for the phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate, alongside some reported experimental values. As shown in Figure 4a, the calculated trend of the phase equilibrium line is almost consistent with the reported experimental values. In the low temperature region of Figure 4a, the phase equilibrium line changes relatively smoothly. As the temperature gradually increases, the phase equilibrium line begins to extend and intersect with the CO₂ liquefaction line (calculated using the method described in Section 2.3.1). In the right region of the intersection point, the trend of the phase equilibrium line changes significantly, and even a slight increase in temperature can lead to a rapid increase in the phase equilibrium pressure, which suggests the difficulties faced by liquid CO₂ in forming hydrates. As shown in Figure 4b, as the NaCl mass fraction increases, the phase equilibrium line shifts to the left, indicating that it is more difficult for CO₂ hydrates to form and easier for them to decompose.

Figure 4. Phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate. (a) Pure water system; (b) aqueous solution system with different NaCl mass fractions. Hollow circle symbols represent experimental [11,16,17,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,44,73] data.

The fitting accuracy of the phase equilibrium conditions of CO₂–pure water/NaCl aqueous solution–CO₂ hydrate is detailed in Table 7. For a pure water system, the three methods proposed in this study exhibit similar effects, and the fitting accuracy reached the level of experimental accuracy. Specifically, the fitting accuracy for method 1 is equal to that of method 3, and both are superior to method 2. For a NaCl aqueous solution system, method 3 performs the best in terms of fitting accuracy, while method 1 and 2 have lower fitting accuracy. The detailed discrepancy statistics across different studies can be found in Table A3.

Table 7. Calculation effect of CO₂–water/NaCl aqueous solution–CO₂ hydrate phase equilibrium conditions.

System	Methods	Fitting Data Points	Fitting Accuracy/%
Pure water with hydrates	1	220	0.243
	2	220	0.245
	3	220	0.243
NaCl aqueous solution with hydrates	1	231	1.190
	2	231	1.095
	3	231	0.210

4. Conclusions

An improved method was proposed to predict CO₂ solubility and phase equilibrium conditions for CO₂–pure water/NaCl aqueous solution mixtures across various temperature and pressure conditions. The core of this method is to modify and combine BWRS EOS with hydrate thermodynamic theories. In the improved method, BWRS EOS was modified by fitting the interaction coefficients based on reported experimental data of CO₂ solubility, which solved the limitation that the conventional BWRS EOS is not suitable for calculating CO₂ solubility, especially in systems involving liquid and hydrate phases. The VdWP model was also modified by fitting the data of phase equilibrium conditions for CO₂–pure water/NaCl aqueous solution–CO₂ hydrate, and three different methods were used to calculate the Langmuir constants.

The prediction accuracy was evaluated by comparison and analysis of the differences between a large number of reported experimental data and the calculation results. The results verified that the fitting accuracy and prediction accuracy matched the experimental standards, and the improved method is useful for the prediction of CO₂ solubility and phase equilibrium conditions for the mixture system with and without CO₂ hydrate presence.

The calculation results highlight the influence of hydrates and NaCl on CO₂ solubility characteristics. In pure water without hydrates, CO₂ solubility increases as the temperature decreases and the pressure increases. Conversely, in pure water with hydrates, it decreases as the temperature decreases and the pressure increases. In NaCl aqueous solution without hydrates, CO₂ solubility increases with decreasing temperature, increasing pressure, and decreasing NaCl mass fraction. In NaCl aqueous solution with hydrates, at high pressure, it increases as the temperature and NaCl mass fraction increase; at low pressure, it decreases with the NaCl mass fraction. Compared to low-pressure systems without hydrates, the impact of pressure is relatively smaller in high-pressure systems with hydrates. Furthermore, the calculated results clearly demonstrate the hindrance of NaCl to the formation of CO₂ hydrates.

This study will be beneficial for the development of CCUS related technologies.